supernumber

Let ΛN be the Grassmann algebra generated by θi, i=1⁢…⁢N,
such that θi⁢θj=-θj⁢θi and (θi)2=0.
Denote by Λ∞, the Grassmann algebra of an infinite number of generatorsθi.
A supernumber is an element of ΛN or Λ∞.

1 Body and soul

The body of a supernumber z is defined as zB=z0,
and its soul is defined as zS=z-zB.
If zB≠0 then z has an inverse given by

z-1=1zB⁢∑k=0∞(-zSzB)k.

2 Odd and even

A supernumber can be decomposed into the even and odd parts:

zeven

=

z0+12⁢zi⁢j⁢θi⁢θj+…+1(2⁢n)!⁢zi1⁢…⁢i2⁢n⁢θi1⁢…⁢θi2⁢n+…,

zodd

=

zi⁢θi+16⁢zi⁢j⁢k⁢θi⁢θj⁢θk+…+1(2⁢n+1)!⁢zi1⁢…⁢i2⁢n+1⁢θi1⁢…⁢θi2⁢n+1+….

Even supernumbers commute with each other and are called c-numbers,
while odd supernumbers anticommute with each other and are called a-numbers.
Note, the product of two c-numbers is even,
the product of a c-number and an a-number is odd,
and the product of two a-numbers is even.
The superalgebra ΛN has the vector spacedecompositionΛN=ℂc⊕ℂa,
where ℂc is the space of c-numbers,
and ℂa is the space of a-numbers.